Because a solution path of a differential equations can not cross
itself, the dynamical in the place can not become too complicated. It is
possible to have equilibrium points or limit cycles. Every orbit which does not
start at an equilibrium point or on limit cicle either converges to one of
these or it escapes to infinity.
A famous problem of David Hilbert
from 1900 asks how many limit cycles a polynomial differential equation can have.
d/dt x=ann xn yn+ ... + a11 xy + a10 x + a01 y + a0
d/dt y=bnn xn yn+ ... + a10 xy + b10 x + b01 y + b0
in the plane can have.
Are there only finitely many limit cycles?
Is there a bound on their number which only depends on n?
A special class of differential equations are Lienard systems
d2/dt2 x + F'(x) d/dt x + G'(x) = 0
which is with y=d/dt x equivalent to
d/dt x = y
d/dt y = -G'(x)-F'(x) y
or with y=d/dx +F, G'(x)=g equivalent to
d/dt x = y-F(x)
d/dt y = -g(x)
Example: van der Pool equation
d/dt x = y
d/dt y = -x -(x2-1) y
Example: Duffing oscillator
d/dt x = y
d/dt y = -x -x3
Last Wednesday,
news
broke that a student Elin Oxenhielm
made progress on the Hilbert problem, claiming that a Lienard system
with g(x)=x and F(x) a polynomial of degree 2k+1 has at most k limit cycles.
The paper containing a sketch of a proof is
here.
Critics doubt that the proof will hold.
